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Mirrors > Home > MPE Home > Th. List > nn0ssz | Structured version Visualization version GIF version |
Description: Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0ssz | ⊢ ℕ0 ⊆ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 12478 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnssz 12585 | . . 3 ⊢ ℕ ⊆ ℤ | |
3 | 0z 12574 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | c0ex 11213 | . . . . 5 ⊢ 0 ∈ V | |
5 | 4 | snss 4789 | . . . 4 ⊢ (0 ∈ ℤ ↔ {0} ⊆ ℤ) |
6 | 3, 5 | mpbi 229 | . . 3 ⊢ {0} ⊆ ℤ |
7 | 2, 6 | unssi 4185 | . 2 ⊢ (ℕ ∪ {0}) ⊆ ℤ |
8 | 1, 7 | eqsstri 4016 | 1 ⊢ ℕ0 ⊆ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∪ cun 3946 ⊆ wss 3948 {csn 4628 0cc0 11114 ℕcn 12217 ℕ0cn0 12477 ℤcz 12563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-i2m1 11182 ax-1ne0 11183 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 |
This theorem is referenced by: nn0z 12588 nn0zd 12589 nn0zi 12592 nn0ssq 12946 nthruz 16201 oddnn02np1 16296 evennn02n 16298 bitsf1ocnv 16390 pclem 16776 0ram 16958 0ram2 16959 0ramcl 16961 gexex 19763 iscmet3lem3 25039 plyeq0lem 25960 dgrlem 25979 2sqreultblem 27188 archirngz 32606 dffltz 41679 diophrw 41800 diophin 41813 diophun 41814 eq0rabdioph 41817 eqrabdioph 41818 rabdiophlem1 41842 diophren 41854 etransclem48 45297 |
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